aa r X i v : . [ h e p - t h ] A ug Tensor Operators in NoncommutativeQuantum Mechanics
Ricardo Amorim
Instituto de F´ısica, Universidade Federal do Rio de Janeiro,Caixa Postal 68528, 21945-970 Rio de Janeiro, Brazil
Abstract
Some consequences of promoting the object of noncommutativ-ity θ ij to an operator in Hilbert space are explored. Its canonicalconjugate momentum is also introduced. Consequently, a consistentalgebra involving the enlarged set of canonical operators is obtained,which permits us to construct theories that are dynamically invariantunder the action of the rotation group. In this framework it is alsopossible to give dynamics to the noncommutativity operator sector,resulting in new features. PACS: 03.65.Fd, 11.10.Nx [email protected] he first published work on space-time noncommutativity appeared in1947 [1], introduced by Snyder as an attempt to avoid singularities in quan-tum field theories; although in recent times this subject has been basicallyrelated to string theory[2].Snyder introduced a five dimensional space-time with SO(4,1) as a sym-metry group, with generators M AB , satisfying the Lorentz algebra. Further-more, he postulated the identification between coordinates and generators ofthe SO (4 ,
1) algebra, x µ = a M µ , promoting in this way the space timecoordinates to hermitian operators. The above identification implies in thecommutation relation [ x µ , x ν ] = ia M µν (1)as well as in following identities: [ M µν , x λ ] = i ( x µ η νλ − x ν η µλ ) and [ M µν , M αβ ] = i ( M µβ η να − M µα η νβ + M να η µβ − M νβ η µα ) , which are in accordance with fourdimensional Lorentz invariance.When open strings have their end points on D-branes in the presence of abackground constant B-field , effective gauge theories on a noncommutativespace-time arise [3, 4]. In these noncommutative field theories (NCFT’s)[5],relation (1) is replaced by [ x µ , x ν ] = iθ µν (2)The point here is that the object of noncommutativity θ µν is usually as-sumed as a constant antisymmetric matrix in NCFT’s. This violates Lorentzsymmetry, but permits to treat NCFT’s as deformations of ordinary quan-tum field theories, replacing in brief ordinary products by Moyal products,and ordinary gauge interactions by the corresponding noncommutative ones.As it is well known, these theories present several deceases as nonunitarity,nonlocalizability, nonrenormalizability, U V x IR mixing etc. At least Lorentzinvariance can be recovered by assuming that θ µν is in fact a tensor operatorwith the same hierarchical level as the x ’s. This was done in [6] by usinga convenient reduction of Snyder’s algebra. As x µ and θ µν belong in thiscase to the same affine algebra, the fields must be functions of the eigenval-ues of both x µ and θ µν . The results appearing in Ref. [6] are explored by A, B = 0 , , , , µ, ν = 0 , , ,
3. The parameter a has dimension of length. Naturalunits are adopted, where ¯ h = c = 1 . x µ , θ αβ ] = 0 (3)The results appearing in [6]-[12] are written in terms of Weyl represen-tations and the related Moyal products. They strongly depend on an in-tegration over parameters related to the objects of noncommutativity, witha weigh function W ( θ ). They use in this process the celebrated Seiberg-Witten[4] transformations.A related subject is given by noncommutative quantum mechanics (NCQM)[14]-[31]. In NCQM, time is kept as a commutative parameter and spacecoordinates do not commute. These assumptions are reasonable in a non-relativistic theory. However most of the authors publishing in the area do notconsider the objects of noncommutativity as Hilbert space operators. Eventhose who consider this possibility do not introduce the corresponding conju-gate momenta, which is necessary to display a complete canonical algebra andto implement rotation as a dynamical symmetry[32]. These facts necessarilyimply that the presented theories fail to be invariant under rotations.In this work we adapt the DFR algebra to non relativistic QM in thesimplest way, but keeping consistency. The objects of noncommutativity areconsidered as true operators and their conjugate momenta are introduced.This permits to display a complete and consistent algebra among the Hilbertspace operators and to construct generalized angular momentum operators,obeying the SO ( D ) algebra, and in a dynamical way, acting properly inall the sectors of the Hilbert space. If this is not done, some fundamentalobjects usually employed in the literature, as the shifted coordinate operator(see (11)) , fail to properly transform under rotations. The symmetry isimplemented not in a mere algebraic way, where the transformations arebased in the indices structure of the variables, but it comes dynamicallyfrom the consistent action of an operator, as discussed in [32].We assume that space has arbitrary D ≥ x i and p i , i =2 , , ...D , represent the position operator and its conjugate momentum. θ ij represent the noncommutativity operator, π ij being its conjugate momentum.In accordance with the discussion above, it follows the algebra[ x i , p j ] = iδ ij , [ θ ij , π kl ] = iδ ijkl (4)where δ ijkl = δ ik δ jl − δ il δ jk . Relation (2) here reads as[ x i , x j ] = iθ ij (5)and the triple commutator condition of the DFR algebra here is written as[ x i , θ jk ] = 0 (6)This implies that [ θ ij , θ kl ] = 0 (7)For simplicity it is assumed that[ p i , θ jk ] = 0 , [ p i , π jk ] = 0 (8)The Jacobi identity formed with the operators x i , x j and π kl leads to thenontrivial relation [[ x i , π kl ] , x j ] − [[ x j , π kl ] , x i ] = − δ ijkl (9)The solution, unless trivial terms, is given by[ x i , π kl ] = − i δ ijkl p j (10)It is simple to verify that the whole set of commutation relations listed aboveis indeed consistent under all possible Jacobi identities. Expression (10)suggests the shifted coordinate operator[15, 17, 25, 26, 29] X i ≡ x i + 12 θ ij p j (11)that commutes with π kl . Actually, (11) also commutes with θ kl and X j , andsatisfies a non trivial commutation relation with p i depending objects, whichcould be derived from 3 X i , p j ] = iδ ij (12)It is possible by now to introduce a continuous basis for a general Hilbertspace, with the aid of the above commutation relations. It is first necessaryto find a maximal set of commuting operators. One can choose, for instance,a momentum basis, formed by the eigenvectors of p , π . A coordinate basisformed by the eigenvectors of X , θ can also be introduced, among otherpossibilities. We observe here that it is in no way possible to form a basisinvolving more than one component of the original position operator x , sincetheir components do not commute.Just for completeness, let us display the fundamental relations involvingthose basis, namely eigenvalue, orthogonality and completeness relations. X i | X ′ , θ ′ > = X ′ i | X ′ , θ ′ > , θ ij | X ′ , θ ′ > = θ ′ ij | X ′ , θ ′ > (13) p i | p ′ , π ′ > = p ′ i | p ′ , π ′ > , π ij | p ′ , π ′ > = π ′ ij | p ′ , π ′ > (14) < X ′ , θ ′ | X ” , θ ” > = δ D ( X ′ − X ”) δ D ( D − ( θ ′ − θ ”) (15) < p ′ , π ′ | p ” , π ” > = δ D ( p ′ − p ”) δ D ( D − ( π ′ − π ”) (16) Z d D X ′ d D ( D − θ ′ | X ′ , θ ′ >< X ′ , θ ′ | = (17) Z d D p ′ d D ( D − π ′ | p ′ , π ′ >< p ′ , π ′ | = (18)Representations of the operators in those bases can be obtained in an usualway. For instance, the commutation relations (4,12) and the eigenvalue rela-tions above, unless trivial terms, yeld < X ′ , θ ′ | p i | X ” , θ ” > = − i ∂∂X ′ i δ D ( X ′ − X ”) δ D ( D − ( θ ′ − θ ”) (19)and < X ′ , θ ′ | π ij | X ” , θ ” > = − iδ D ( X ′ − X ”) ∂∂θ ′ ij δ D ( D − ( θ ′ − θ ”) (20)4he transformations from one basis to the other are done by extended Fouriertransforms. Related with these transformations is the ”plane wave” < X ′ , θ ′ | p ” , π ” > = N exp( ip ” X ′ + iπ ” θ ′ ), where internal products are assumed, from now, in thepertinent expressions. For instance, p ” X ′ + π ” θ ′ = p ” i X ′ i + π ” ij θ ′ ij .Before discussing any dynamics, it seems interesting to study the gener-ators of the group of rotations SO ( D ). Not considering the spin sector, wesee that the usual angular momentum operator l ij = x i p j − x j p i (21)does not close in an algebra due to (5). Actually[ l ij , l kl ] = iδ il l kj − iδ jl l ki − iδ ik l lj + iδ jk l li + iθ il p k p j − iθ jl p k p i − iθ ik p l p j + iθ jk p l p i (22)and so their components can not be SO ( D ) generators in this extendedHilbert space. It is not hard to see that, on the contrary, the operator L ij = X i p j − X j p i (23)closes in the SO ( D ) algebra. However, to properly act in the θ, π sector, ithas to be generalized to the total angular momentum operator J ij = L ij − θ il π jl + θ jl π il (24)As can be verified, not only[ J ij , J kl ] = iδ il J kj − iδ jl J ki − iδ ik J lj + iδ jk J li (25)but J ij generates rotations in all of the Hilbert space sectors. Actually δ X i = i ǫ kl [ X i , J kl ] = ǫ ik X k δ p i = i ǫ kl [ p i , J kl ] = ǫ ik p k δθ ij = i ǫ kl [ θ ij , J kl ] = ǫ ik θ jk + ǫ jk θ ik δπ ij = i ǫ kl [ p ij , J kl ] = ǫ ik π jk + ǫ jk π ik (26)5ave the expected form. The same occurs with x i = X i − θ ij p j : δ x i = i ǫ kl [ x i , J kl ] = ǫ ik x k . Observe that in the usual NCQM prescription, wherethe objects of noncommutativity are parameters or where the angular mo-mentum operator has not been generalized, X fails to transform as a vectoroperator under SO ( D )[15, 17, 25, 26, 29]. The consistence of transforma-tions (26) comes from the fact that they are generated through the action ofa symmetry operator and not from operations based on the index structureof those variables.We would like to mention that in D = 2 the operator J ij reduces to L ij ,in accordance with the fact that in this case θ or π has only one independentcomponent. In D = 3, it is possible to represent θ or π by three vectorsand both parts of the angular momentum operator have the same kind ofstructure, and so the same spectrum. An unexpected addition of angularmomentum potentially arises, although the θ, π sector can leave in a j = 0Hilbert subspace. Unitary rotations are generated by U ( ω ) = exp( − iω. J ),while unitary translations, by T ( λ, Ξ) = exp( − iλ. p − i Ξ .π ).To close this work, let us consider the isotropic D-dimensional harmonicoscillator. There are several possibilities of rotational invariant Hamiltonianswhich present the proper commutative limit[17, 18, 26, 27]. A simple one isgiven by H = 12 m p + mω X (27)since X i commutes with X j , satisfies the canonical relation (12) and in thepresent formalism transforms according to (26). This permits to constructannihilation and creation operators in the usual way: A i = q mω ( X i + i p i mω )and A † i = q mω ( X i − i p i mω ). They satisfy the usual harmonic oscillator al-gebra, and H can be written in terms of the sum of D number operators N i = A † i A i , presenting the same spectrum and the same degeneracies whencompared with the ordinary QM case [33]. The θ, π sector, however, is notcontemplated with any dynamics if H represents the total Hamiltonian. Asthe harmonic oscillator describes a system near an equilibrium configuration,it seems interesting as well to add to (27) a new term like H θ = 12Λ π + ΛΩ θ (28)6here Λ is a parameter with dimension of ( lengh ) − and Ω is some fre-quency. Both Hamiltonians can be simultaneously diagonalized, since theycommute. So the total Hamiltonian eigenstates will be formed by the directproduct of the Hamiltonian eigenstates of each sector. Let us consider the θ, π sector. Annihilation and creation operators are respectively defined as A ij = q ΛΩ2 ( θ ij + iπ ij ΛΩ ) and A † ij = q ΛΩ2 ( θ ij − iπ ij ΛΩ ). They satisfy the oscillatoralgebra [ A ij , A † kl ] = δ ij,kl permitting to construct eigenstates of H θ associ-ated with quantum numbers n ij . As usual, the ground state is annihilatedby A ij , and its corresponding wave function ( in the θ, π sector ) is < θ ′ | n ij = 0 , t > = ( ΛΩ π ) D ( D − exp [ − ΛΩ4 θ ′ ij θ ′ ij ] exp [ − iD ( D −
1) Ω4 t ] (29)The wave functions for excited states are obtained through the applicationof the creation operator A † kl on the fundamental state. However, we expectthat Ω might be so big that only the fundamental level of this generalizedoscillator could be populated. This will generate only a shift in the oscillatorspectrum, valuing ∆ E = D ( D − Ω. This fact seems to be trivial, but this newvacuum energy could generate unexpected behaviors. Another point relatedwith (29) is that it gives a natural way for introducing the weight function W ( θ ) which appears, in the context of NCFT’s, in Refs. [6, 7]. Actually, justconsidering the θ, π sector, the expectation value of any function f ( θ ) overthe fundamental state is < f ( θ ) > = < n kl = 0 , t | f ( θ ) | n kl = 0 , t > = ( ΛΩ π ) D ( D − Z d D ( D − θ ′ f ( θ ′ ) exp [ − ΛΩ2 θ ′ rs θ ′ rs ] ≡ Z d D ( D − θ ′ W ( θ ′ ) f ( θ ′ ) (30)where W ( θ ′ ) ≡ ( ΛΩ π ) D ( D − exp [ − ΛΩ2 θ ′ rs θ ′ rs ] (31)giving the expectation values 7 > = 1 < θ ij > = 012 < θ ij θ ij > = < θ >< θ ij θ kl > = 2 D ( D − δ ij,kl < θ > (32)with < θ > ≡
12 Λ Ω .This result permits to calculate expectation values of the physical co-ordinate operators. As one can verify, < x i > = < X i > = 0, but one canfind non trivial noncommutativity contributions to the expectation valuesof other operators. For instance, it is easy to see from (32) and (11) that < x > = < X > + D < θ >< p > , where < X > and < p > are theusual QM results for an isotropic oscillator in a given state. This shows thatnoncommutativity enlarges the root-mean-square deviation of the physicalcoordinate operator, as expected. This is an important result, which couldbe in principle measurable. The inclusion of gauge interactions, the super-symmetrization and a possible relativistic generalization of this theory areunder consideration and will be published elsewhere. Acknowledgment:
I am indebted to the UFRJ Group of Casimir Effect forimportant discussions. This work is partially supported by CNPq and FUJB(Brazilian Research Agencies).
References [1] H. S. Snyder, Phys. Rev. (1947) 38.[2] See for instace J. Polchinski, String Theory, University Press, Cam-bridge , 1998; R. Szabo, An introduction to String Theory and D-BraneDynamics, Imperial College Press, London, 2004.[3] M.R.Douglas and C. Hull, JHEP (1998) 008; M. M. Sheikh-Jabbari, Phys. Lett B 450 (1999) 119.[4] N. Seiberg and E. Witten, JHEP (1999) 032.85] See, for instance, R. J. Szabo, Phys. Repp (2003) 207.[6] C. E. Carlson, C.D. Carone and N. Zobin, Phys. Rev.
D 66 (2002)075001.[7] H. Kase, K. Morita, Y. Okumura and E. Umezawa, Prog. Theor. Phys. (2003) 663; K. Imai, K. Morita and Y. Okumura, Prog. Theor.Phys. (2203) 989.[8] R. Banerjee, B. Chakraborty and K. Kumar, Phys. Rev.
D 70 (2004)125004.[9] M. Haghighat and M. M. Ettefaghi, Phys. Rev
D 70 (2004) 034017.[10] C. D. Carone and H. J. Kwee, Phys. Rev.
D 73 (2006) 096005.[11] M. M. Ettefaghi and M. Haghighat, Phys. Rev
D 75 (2007) 125002.[12] S. Saxell
On general properties of Lorentz invariant formulation of non-commutative quantum field thery , hep-th 08043341.[13] S. Doplicher, K. Fredenhagen and J. E. Roberts, Phys. Lett.
B331 (1994) 29; Commun. Math. Phys. (1995) 187.[14] C. Durval and P. Horvathy, Phys. Lett.
B 479 (2000) 284[15] M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Phys. Rev. Lett (2001) 2716.[16] M. Chaichian, A. Demichec, P. Presnajder, M. M. Sheikh-Jabbari andA. Tureanu, Nucl. Phys. B 611 (2002) 383.[17] J. Gamboa, M. Loewe and J. C. Rojas, Phys. Rev.
D 64 (2001) 067901.[18] V. P .Nair and A. P. Polychronakos, Phys. Lett
B 505 (2001) 267.[19] R. Banerjee, Mod. Phys. Lett. (2002) 631.[20] Stefano Bellucci and A. Nersessian, Phys. Lett. B 542 (2002) 295.[21] P.-M. Ho and H.-C. Kao, Phys. Rev Lett (2002) 151602.922] A. A. Deriglazov, Phys. Lett. B555 (2003) 83; JHEP (2003) 021.[23] A. Smailagic and E. Spallucci, J. Phys.
A36 (2003) L467; J. Phys.
A36 (2003) L517.[24] L. Jonke and S. Meljanac, Eur. Phys. Jour.
C29 (2003) 433.[25] A. Kokado, T. Okamura and T. Saito, Phys.
D 69 (2004) 125007.[26] A. Kijanka and P Kosinski, Phys. Rev.
D 70 (2004) 127702.[27] I. Dadic, L. Jonke and S. Meljanac, Acta Phys. Slov. (2005) 149.[28] S. Bellucci and A. Yeranyan, Phys. Lett. B 609 (2005) 418.[29] X. Calmet, Phys. Rev.
D 71 (2005) 085012; X. Calmet and M. Selvaggi,Phys. Rev
D74 (2006) 037901.[30] F. G. Scholtz, B. Chakraborty, J. Govaerts and S. Vaidya, J. Phys.
A40 (2007) 14581.[31] M. Rosenbaum, J. David Vergara and L R. Juarez, Phys. Lett.
A 367 (2007) 1.[32] A. Iorio, Phys. Rev.